Generalized Space–Time Fractional Dynamics in Networks and Lattices

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Generalized Space–Time Fractional Dynamics in Networks and Lattices

Thomas Michelitsch, Alejandro Perez Riascos, Bernard Collet, Andrzej F.

Nowakowski, Franck Nicolleau

To cite this version:

Thomas Michelitsch, Alejandro Perez Riascos, Bernard Collet, Andrzej F. Nowakowski, Franck Nicol-

leau. Generalized Space–Time Fractional Dynamics in Networks and Lattices. H Altenbach, V Ere-

meyev, I Pavlov, A. Porubov. Nonlinear Wave Dynamics of Materials and Structures. Advanced

Structured Materials, 122, Springer, Cam, pp.221-249, 2020, Advanced Structured Materials, 978-3-

030-38707-5. �10.1007/978-3-030-38708-2_14�. �hal-02314815�

Generalized space-time fractional dynamics in networks and lattices

T.M. Michelitsch

^1∗

, A.P. Riascos

²

, B.A. Collet

¹

, A.F. Nowakowski

³

, F.C.G.A. Nicolleau

³

1

Sorbonne Universit´e

Institut Jean le Rond d’Alembert, CNRS UMR 7190 4 place Jussieu, 75252 Paris cedex 05, France

2

Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 20-364, 01000 Ciudad de M´exico, M´exico

3

Department of Mechanical Engineering University of Sheffield

Mappin Street, Sheffield S1 3JD, United Kingdom October 13, 2019

Abstract

We analyze generalized space-time fractional motions on undirected networks and lattices. The continuous- time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non- Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density.

We analyze ‘generalized space-time fractional diffusion’ in the infinite d-dimensional integer lattice Z^d. We obtain in the diffusion limit a ‘macroscopic’ space-time fractional diffusion equation. Classical CTRW models such as with Laskin’s fractional Poisson process and standard Poisson process which occur as spe- cial cases are also analyzed. The developed generalized space-time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems.

1

INTRODUCTION

Random walk models are considered to be the most fundamental approaches to describe stochastic processes in nature. Hence applications of random walks cover a very wide area in fields as various as random search strategies, the proliferation of plant seeds, the spreading phenomena of pandemics or pollution, chemical reactions, finance, population dynamics, properties of public transportation networks, anomalous diffusion and generally such approaches are able to capture empirically observed power-law features in ‘complex systems’

[1, 2, 3, 4, 5, 6, 7]. On the other hand the emergence of “network science”, and especially the study of random walks on networks has become a major subject for the description of dynamical properties in complex systems [8, 9, 10].

In classical random walks in networks, the so-called ‘normal random walks’, the walker in one step can reach only connected next neighbor sites [11, 12]. To the class of classical Markovian walks refer continuous- time random walks (CTRWs) where the walker undertakes jumps from one node to another where the waiting time between successive jumps are exponentially distributed leading to Poisson distributed numbers of jumps.

The classical CTRW models with Poisson renewal process are able to capture normal diffusive properties such as the linear increase of the variance of a diffusing particle [6]. However, these classical walks are unable to describe power-law features exhibited by many complex systems such as the sublinear time characteristics

∗Corresponding author, e-mail : michel@lmm.jussieu.fr

of the mean-square displacement in anomalous diffusion [5]. It has been demonstrated that such anomalous diffusive behavior is well described by a random walk subordinated to the fractional generalization of the Poisson process. This process which was to our knowledge first introduced by Repin and Saichev [13], was developed and analyzed by Laskin who called this process the ‘fractional Poisson process’ [14, 15]. The Laskin’s fractional Poisson process was further generalized in order to obtain greater flexibility to adopt real-world situations [16, 17, 18]. We refer this renewal process to as ‘generalized fractional Poisson process’ (GFPP).

Recently we developed a CTRW model of a normal random walk subordinated to a GFPP [17, 18].

The purpose of the present paper is to explore space fractional random walks that are subordinated to a GFPP. We analyze such motions in undirected networks and as a special application in the multidimensional infinite integer lattice

Z^d

.

2

RENEWAL PROCESS AND CONTINUOUS-TIME RANDOM WALK

In the present section, our aim is to give a brief outline of renewal processes (or also referred to as ‘compound processes’) and closely related to the ‘continuous-time random walk (CTRW)’ approach which was introduced by Montroll and Weiss [19]. For further outline of renewal theory and related subjects we refer to the references [2, 4, 19, 20, 21, 22]. It is mention worthy that we deal in this paper with causal generalized functions and distributions in the sense of Gelfand and Shilov [23].

We consider a sequence of randomly occurring ‘events’. Such events can be for instance the jumps of a diffusing particle or failure events in technical systems. We assume that the events occur at non-negative random times 0 ≤ t

1

, t

2

, . . . t

n

, . . . , ∞ where t = 0 represents the start of the observation. The random times t

k

when events occur are called ‘arrival times’. The time intervals between successive events ∆t

_k

= t

_k

− t

k−1

≥ 0 are called ‘waiting times’ or ‘interarrival times’ [20]. The random event stream is referred to as a renewal process if the waiting time ∆t between successive events is an ‘independent and identically distributed’ (IID) random variable which may take any non-negative continuous value. This means in a renewal process the waiting time ∆t

_k

between successive events is drawn ∀ k from the same waiting probability density function (PDF) χ(t ). This distribution function is called waiting time distribution function (waiting time PDF) or short waiting time density

¹

. The quantity χ(t )dt indicates the probability that an event occurs at time t (within [t , t + dt ]).

We can then write the probability Ψ(t ) that the waiting time for the first event is ∆t ≤ t or equivalently that at least one event occurs in the time interval [0 , t ] as

Ψ(t ) =

Z t

0

χ(τ )dτ, t ≥ 0 , lim

t→∞

Ψ(t ) = 1 − 0 (1)

with the obvious initial condition Ψ (t = 0 ) = 0 . The distribution (1) in the context of lifetime models often is also called ‘failure probability’ [20]. From this relation follows that the waiting time density χ(t ) is a normalized PDF. In classical renewal theory the waiting time PDF was assumed to be exponential χ(t ) = ξe

^−ξt

(ξ > 0 ) which leads as we will see later to Markovian memoryless Poisson type processes.

The waiting time PDF has physical dimension sec

⁻¹

and the cumulative distribution (1) indicates a dimensionless probability. Another quantity of interest is the so called ‘survival probability’ Φ

⁽⁰⁾

(t ) defined as

Φ

⁽⁰⁾

(t ) = 1 − Ψ(t ) =

Z ∞

t

χ(τ )dτ (2)

which indicates the (dimensionless) probability that no event has occurred within [0 , t ], i.e. in a random walk the probability that the walker at time t still is waiting on its departure site. Of further interest is the PDF of the arrival of n jump events which we denote by χ

⁽ⁿ⁾

(t ) (χ

⁽ⁿ⁾

(t )dt being the probability that the nth jump is performed at time t ). Since the events are IID we can establish the recursion

χ

⁽ⁿ⁾

(t ) =

Z t

0

χ

⁽ⁿ⁻¹⁾

(τ )χ(t − τ )dτ, χ

⁽⁰⁾

(t ) = δ(t ) (3)

1In the context of random walks where the events indicate random jumps we also utilize the notion ‘jump density’ [17].

and with χ

⁽¹⁾

(t ) = χ(t ). Thus the PDF for the arrival of the n th event is given by the n − 1 fold convolution of χ(t ) with itself, namely

χ

⁽ⁿ⁾

(t ) =

Z ∞

0

. . .

Z ∞

0

χ(τ

1

) . . . χ(τ

n

)δ



t

−

n

X

j=1

τ

j





dτ

1

. . . dτ

n

, t > 0 , n = 1 , 2 , . . . . (4) In this relation we have assumed that the waiting time PDF is causal, i.e. χ(t ) is non-zero only for t ≥ 0 . For an outline of causal distributions and some of their properties especially Laplace transforms, see Appendix A.1. The probability that n events happen within time interval [0 , t ] then can be written as

Φ

⁽ⁿ⁾

(t) =

Z t

0

(1 − Ψ(t − τ )) χ

⁽ⁿ⁾

(τ )dτ =

Z t

0

Φ

⁽⁰⁾

(t − τ )χ

⁽ⁿ⁾

(τ )dτ, n = 0, 1, 2, . . .

=

Z t

0

Φ

⁽ⁿ⁻¹⁾

(t − τ )χ(τ )dτ.

(5)

This convolution takes into account that the nth event may happen at a time τ < t and no further event is taking place during t − τ with survival probability Φ

⁽⁰⁾

(t ) where 0 ≤ τ ≤ t . The distribution Φ

⁽ⁿ⁾

(t ) are dimensionless probabilities whereas the PDFs χ

⁽ⁿ⁾

(t ) have physical dimension of sec

⁻¹

. It is especially instructive to consider all these convolution relations in the Laplace domain. We then obtain with (4) the convolution relation

˜

χ

⁽ⁿ⁾

(s) =

Z ∞

0

. . .

Z ∞

0

χ(τ

1

) . . . χ(τ

n

)e

^−st

δ



t

−

n

X

j=1

τ

j





dτ

1

. . . dτ

n

=

Z ∞

0

χ(t)e

^−st

dt

n

= ( ˜ χ(s))

ⁿ

, n = 0, 1, 2, . . .

(6)

where χ ˜

⁽⁰⁾

(s) = 1 indeed recovers χ

⁽⁰⁾

(t ) = δ(t ) for n = 0 . This relation also shows that the density of n events χ

⁽ⁿ⁾

(t ) is normalized, namely

˜

χ

⁽ⁿ⁾

(s) |

s=0

= 1 (7)

as a consequence of the normalization of the waiting time PDF χ(t ). Now in view of (1) and (2) it is straightforward to obtain the Laplace transforms

Ψ(s) = ˜ χ(s) ˜

s , Φ ˜

⁽⁰⁾

(s) = 1

s − Ψ(s) = ˜ 1 − χ(s) ˜

s (8)

thus the Laplace transform of the probability distribution (5) for n events is given by Φ ˜

⁽ⁿ⁾

(s) = Φ ˜

⁽⁰⁾

(s)(˜ χ(s))

ⁿ

= 1 − ˜ χ(s)

s ( χ(s)) ˜

ⁿ

, n = 0 , 1 , 2 , . . . (9)

For a brief demonstration of further general properties of renewal processes it is convenient to introduce the following generating function

G(t , v) =

∞

X

n=0

v

ⁿ

Φ

⁽ⁿ⁾

(t ) (10)

and its Laplace transform

²

G(s, ˜ v) = L{ G(t , v ) } =

∞

X

n=0

v

ⁿ

Φ ˜

⁽ⁿ⁾

(s). (11)

2We denote˜f(s) =L{f(t)}the Laplace transform off(t) and byL⁻¹{. . .} Laplace inversion, see Appendix A.1 for further details.

Taking into account (9) together with the obvious property | χ(s) ˜ | ≤ | χ(s ˜ = 0 ) | = 1 we get for (11) a geometric series

G(s, ˜ v) = 1 − χ(s) ˜ s

∞

X

n=0

v

ⁿ

(˜ χ(s ))

ⁿ

= 1 − χ(s) ˜ s

1

1 − v ˜ χ(s) (12)

converging for | v(˜ χ(s)) | < 1 , i.e. for | v | ≤ 1 if s 6 = 0 and | v | < 1 for s = 0 . We directly observe in this relation the normalization condition

L

⁻¹

{ G(s, ˜ v) |

v=1

} = L

⁻¹

1

s

= G(t , 1 ) =

∞

X

n=0

Φ

⁽ⁿ⁾

(t ) = 1 , t > 0 . (13) The generating function is often useful for the explicit determination of the Φ

⁽ⁿ⁾

(t ), namely

Φ

⁽ⁿ⁾

(t ) = 1 n!

d

ⁿ

dv

ⁿ

G(t , v)

v=0

= L

⁻¹

1 − χ(s) ˜

s (˜ χ(s))

ⁿ

(14) where the Laplace transform of this relation recovers by accounting for (12) again the expression (9).

Of further interest is the expected number of events ¯ n(t ) that are taking place within the time interval [0 , t ]. This quantity can be obtained from the relation

¯ n (t ) =

∞

X

n=0

nΦ

⁽ⁿ⁾

(t ) = d

dv G(t , v)

v=1

= L

⁻¹

d

dv

G(s, ˜ v)

v=1

= L

⁻¹

χ(s) ˜ s(1 − χ(s)) ˜

. (15)

2.1 POISSON PROCESS

Before we pass on to non-classical generalizations, it appears instructive to recall some properties of the classical variant which is the ‘Poisson renewal process’ (compound Poisson process) [24]. In this process the waiting time PDF has exponential form

χ

_P

(t ) = ξe

^−ξt

Θ(t ), ξ > 0 (16)

where ξ is a characteristic constant with physical dimension sec

⁻¹

where ξ

⁻¹

defines a characteristic time scale in the process. With the Heaviside Θ (t )-function we indicate here that (16) is a causal distribution

³

. We see that (16) is a normalized PDF which has the Laplace transform

˜

χ

P

(s) = ξ

Z ∞

0

e

^−st

e

^−ξt

dt = ξ

ξ + s (17)

where χ ˜

P

(s = 0 ) = 1 reflects normalization of waiting time PDF (16). Then we get straightforwardly the failure and survival probabilities, respectively

Ψ

_P

(t ) = 1 − e

^−ξt

, Φ

_P⁽⁰⁾

(t ) = 1 − Ψ

_P

(t ) = e

^−ξt

. (18) Also the generating function can be written down directly as

G ˜

P

(s, v ) = 1 ξ + s

∞

X

n=0

(ξv)

ⁿ

(ξ + s)

ⁿ

= 1

ξ(1 − v) + s <{ s } > ξ, (19) thus

G

P

(t , v ) = e

^{−ξ(1−v)t}

. (20)

By using (14) we obtain then for the probability of n events Φ

_P⁽ⁿ⁾

(t ) = 1

n ! d

ⁿ

dv

ⁿ

e

^(v−1)ξt

v=0

= (ξt )

ⁿ

n! e

^−ξt

, n = 0 , 1 , 2 , . . . , t ≥ 0 (21)

3We often skipΘ(t) when there is no time derivative involved.

which is the Poisson distribution. Therefore the renewal process generated by an IID exponential waiting time PDF (16) is referred to as Poisson renewal process or also compound Poisson process. This process is the classical proto-example of renewal process [20, 24] (and see the references therein). We further mention in view of Eq. (15) that the average number of events n ¯ (t ) taking place within [0 , t ] is obtained as

¯

n

p

(t ) = d

dv G

P

(t , v )

v=1

= d

dv e

^(v−1)ξt

v=1

= ξt , t ≥ 0 . (22) In a Poisson renewal process the expected number of arrivals increases linearly in time. The exponential decay in the distributions related to the Poisson process make this process memoryless with the Markovian property [17, 20].

3

FRACTIONAL POISSON PROCESS

In anomalous diffusion one has for the average number of arrivals instead of the linear behavior (22) a power law ∼ t

^β

with 0 < β < 1 [5, 17, 18], among others. To describe such anomalous power-law behavior a

‘fractional generalization’ of the classical Poisson renewal process was introduced and analyzed by Laskin [14, 15] and others [13, 20, 25]. The fractional Poisson renewal process can be defined by a waiting time PDF with the Laplace transform

˜

χ

_β

(s) = ξ

s

^β

+ ξ , ξ > 0 , 0 < β ≤ 1 . (23)

The fractional Poisson process introduces long-time memory effects with non-Markovian features. We will come back to these issues later. The constant ξ has here physical dimension sec

^−β

defining a characteristic time scale in the fractional Poisson process. For β = 1 the fractional Poisson process recovers the standard Poisson process outlined in the previous section. The waiting time density of the fractional Poisson process is then defined by

χ

_β

(t ) = L

⁻¹

ξ

s

^β

+ ξ

= L

⁻¹

ξs

^−β

1

1 + ξs

^−β

. (24)

In order to evaluate the inverse Laplace transform it useful to expand (1 + ξs

^−β

)

⁻¹

into a geometric series with respect to ξs

^−β

which converges for s = σ + iω with σ = <{ s } > ξ

^1β

for all ω. Doing so we obtain

χ

_β

(t ) =

∞

X

m=0

( − 1 )

^m

ξ

^m+1

L

⁻¹

{ s

^−β(m+1)

} , 0 < β ≤ 1 , <{ s } > ξ

^1β

. (25) Taking into account the inverse Laplace transform

⁴

L

⁻¹

{ s

^−µ

} = Θ(t )

^t_Γ(µ)^µ−1

where µ > 0 (See also Appendix A.1 for the discussion of some properties). We obtain then for (25) [14, 17, 20]

χ

_β

(t ) = ξt

^β−1

∞

X

m=0

( − ξt

^β

)

^m

Γ (βm + β) , 0 < β ≤ 1 , t > 0

= ξt

^β−1

E

_β,β

( − ξt

^β

) = d

dt (1 − E

_β

( − ξt

^β

))

(26)

where in this relation we introduced the generalized Mittag-Leffler function E

β,γ

(z ) and the standard Mittag- Leffler function E

_β

(z ) defined in the Appendix A.1 by Eqs. (99) and (100), respectively. The waiting time PDF of the fractional Poisson process also is referred to as Mittag-Leffler density and was introduced first by Hilfer and Anton [26]. It is now straightforward to obtain in the same way the survival probability for the fractional Poisson process, namely (See also Eq. (8))

Φ

_β⁽⁰⁾

(t ) = L

⁻¹

s

^β−1

s

^β

+ ξ

= E

β

( − ξt

^β

), 0 < β ≤ 1 . (27)

4In this relation in the sense of generalized functions we can include the value µ=0 as as he limit limµ→0+t^µ−1

Γ(µ) =δ(t) [23].

The generating function (10) is then by accounting for (27) obtained as G

β

(t , v ) = L

⁻¹

s

^β−1

ξ(1 − v ) + s

^β

= E

β

( − ξ(1 − v)t

^β

), t ≥ 0 . (28) For v = 1 this relation takes G(t , 1 ) = Θ(t ) = 1 (t ≥ 0 ) and for β = 1 the Poisson exponential (20) is recovered. The probability for n arrivals within [0 , t ] is then with relation (14) obtained as

Φ

_β⁽ⁿ⁾

(t ) = 1 n!

d

ⁿ

dv

ⁿ

E

_β

((v − 1 )ξt

^β

)

v=0

= (ξt

^β

)

ⁿ

n!

∞

X

m=0

(m + n )!

m!

( − ξt

^β

)

^m

Γ (β(m + n ) + 1 ) , 0 < β ≤ 1 . (29) This distribution is called the fractional Poisson distribution and is of utmost importance in fractional dy- namics, generalizing the Poisson distribution (21) [14, 15]. For β = 1 the fractional Poisson distribution (29) turns into the classical Poisson distribution (21). We directly confirm the normalization of the fractional Poisson distribution by the relation

∞

X

n=0

Φ

_β⁽ⁿ⁾

(t ) =

∞

X

n=0

1 n !

d

ⁿ

dv

ⁿ

E

_β

((v − 1 )ξt

^β

)

v=0

= E

_β

(( − 1 + 1 )ξt

^β

) = E

_β

(0 ) = 1 . (30) We notice that for β = 1 the Mittag-Leffler function becomes the exponential E

1

( − ξt ) = e

^−ξt

thus the distributions of the standard Poisson process of last section are then reproduced. It is worthy to consider the distinct behavior of the fractional Poisson process for large observation times. To this end let us expand Laplace transform (23) for | s | small which governs the asymptotic behavior for large times

˜ χ

_β

(s) =

1 + s

^β

ξ

−1

=

∞

X

m=0

( − 1 )

^m

ξ

^(−m)

s

^mβ

= 1 − 1

ξ s

^β

+ . . . (31)

which yields as asymptotically for large observation times for 0 < β < 1 , α > 0 fat-tailed behavior

⁵

χ

_β

(t ) ≈ − 1

ξΓ ( − β) t

^−β−1

, 0 < β < 1 , α > 0 , t → ∞ . (32) The fat-tailed behavior χ

β

(t ) ∼ t

^−β−1

is a characteristic power-law feature of the fractional Poisson renewal process reflecting the non-locality in time that produces Laplace transform (31) within the fractional index range 0 < β < 1 . As a consequence of the fat-tailed behavior for 0 < β < 1 extremely long waiting times occur thus the fractional Poisson process is non-Markovian exhibiting long-time memory effects [17, 20].

Further of interest is the power-law tail in the fractional Poisson distribution. We obtain this behavior by considering the lowest power in their Laplace transform, namely

Φ

_β⁽ⁿ⁾

(t ) = L

⁻¹

1

s

(1 + ξ

⁻¹

s

^β

)

⁻ⁿ

− (1 + ξ

⁻¹

s

^β

)

⁻ⁿ⁻¹

≈ L

⁻¹

s

^β−1

ξ

≈ (t ξ

^1β

)

^−β

Γ (1 − β) 0 < β < 1 , n = 0 , 1 , . . . , t ξ

^1β

→ ∞ .

(33)

The fractional Poisson distribution exhibits for large (dimensionless) observation times t ξ

^1β

→ ∞ universal t

^−β

power-law behavior independent of the arrival number n. We will come back subsequently to this important issue.

5Note that−Γ(−β) =β⁻¹Γ(1−β)>0.

4

GENERALIZATION OF THE FRACTIONAL POISSON PROCESS

In this section our aim is to develop a renewal process which is a generalization of the fractional Poisson process of previous section. The waiting time PDF of this process has the Laplace transform

˜

χ

β,α

(s) = ξ

^α

(s

^β

+ ξ)

^α

, 0 < β ≤ 1 , α > 0 , ξ > 0 . (34)

This process was first introduced by Cahoy and Polito [16]. We referred the renewal process defined by (34) to as the generalized fractional Poisson process (GFPP) [17, 18]. The characteristic dimensional constant ξ in (34) has as in the fractional Poisson process physical dimension sec

^−β

and defines a characteristic time scale.

The GFPP contains further two index parameters α > 0 and 0 < β ≤ 1 . The advantage of generalizations such as the GFPP is that they offer a larger parameter space allowing to greater flexibility in adapting to real-world situations. The GFPP recovers for α = 1 , 0 < β < 1 the above described fractional Poisson process and for α = 1 , β = 1 the standard Poisson process and for β = 1 , α > 0 the so called (generalized) Erlang process where α is allowed to take positive integer or non-integer values [17]. The waiting time density of the GFPP is then obtained as (See also Ref. [16])

χ

_β,α

(t ) = ξ

^α

∞

X

m=0

( − 1 )

^m

(α)

_m

m ! ξ

^m

L

⁻¹

{ s

^−β(m+α)

} , t > 0 , σ = <{ s } > ξ

^1β

, 0 < β ≤ 1 , α > 0

= ξ

^α

t

^βα−1

∞

X

m=0

(α)

m

m!

( − ξt

^β

)

^m

Γ (βm + αβ) = ξ

^α

t

^βα−1

E

_β,αβ^α

( − ξt

^β

).

(35)

In this expression is introduced a generalization of the Mittag-Leffler function which was first described by Prabhakar [27] and is defined by

E

_a,b^c

(z ) =

∞

X

m=0

(c)

m

m !

z

^m

Γ (am + b) , <{ a } > 0 , <{ b } > 0 , c, z ∈

C.

(36) In the Prabhakar-Mittag-Leffler function (36) and in the expansion (35) we introduced the Pochhammer symbol (c)

_m

which is defined as [28]

(c)

m

= Γ (c + m) Γ (c) =







1, m = 0

c(c + 1) . . . (c + m − 1), m = 1, 2, . . .

(37) Despite Γ (c) is singular at c = 0 the Pochhammer symbol can be defined also for c = 0 by the limit (0 )

m

= lim

c→0+

(c)

m

= δ

m0

which is also fulfilled by the right-hand side of (37). Then (c)

m

is defined for all c ∈

C

thus we have E

_a,b⁰

(z ) = 1 . The series (36) converges absolutely in the entire complex z - plane. The Prabhakar-Mittag-Leffler function (36) and related problems were analyzed by several authors [28, 29, 30, 31, 32].

In Figure 1(a) is drawn the waiting time PDF of Eq. (35) for a fixed value of α = 2 and variable β in the admissible range 0 < β ≤ 1 . The waiting time PDF exhibits for t small the power-law behavior χ

_β,α

(t ) ≈

_Γ(αβ)^ξα

t

^αβ−1

(corresponding to the zero order in the expansion (35)) with two distinct regimes: For αβ < 1 the waiting time PDF becomes singular at t = 0 corresponding to ‘immediate’ arrivals of the first event. For αβ = 1 the jump density takes the constant value χ

_α⁻¹_,α

(t = 0 ) = ξ

^α

whereas for αβ > 0 the waiting time density χ

β,α

(t = 0 ) = 0 tends to zero as t → 0 where the waiting times become longer the larger αβ.

In Figure 1(b) we depict the behavior of the waiting time PDF for fixed β = 0 .5 and 0 < α ≤ 2 thus

αβ ≤ 1 . It can be seen that the smaller αβ the more narrowly the waiting time PDF is concentrated at small

t -values close to t = 0 . This behavior can also be identified in view of Laplace transform (34) which takes in

0.0 0.2 0.4 0.6 0.8 1.0

t

10⁻² 10⁻¹ 10⁰ 10¹

χ β ,α ( t )

(a) α = 2.0

0.00 0.25 0.50 0.75 1.00

β

0.0 0.2 0.4 0.6 0.8 1.0

t

10⁻² 10⁻¹ 10⁰ 10¹

χ β ,α ( t )

(b) β = 0.5

0.0 0.5 1.0 1.5 2.0

α

Figure 1: The waiting time density density χ

_β,α

(t) as a function of t. We explore the results for (a) α = 2.0 for different values 0 < β ≤ 1 and (b) β = 0.5 for 0 < α ≤ 2 (in each case, the parameters are codified in the colorbar). Results were obtained numerically using ξ = 1 and Eqs. (35) and (36). We depict with dashed lines the case when αβ = 1.

the limit α → 0 the value lim

α→0

χ ˜

_β,α

(s) = 1 thus χ

_β,0+

(t ) = L

⁻¹

(1 ) = lim

α→0 ξ^α

Γ(αβ)

t

^αβ−1

= δ(t ) exhibits the shape of a Dirac δ-distribution peak.

Now our goal is to determine the generalization of the fractional Poisson distribution (29) which is deter- mined by Eq. (9) with (34), namely

Φ

_β,α⁽ⁿ⁾

(t ) = L

⁻¹

1

s

˜

χ

ⁿ_β,α

(s) − χ ˜

ⁿ⁺¹_β,α

(s)

= L

⁻¹

1

s χ ˜

_β,nα

(s) − χ ˜

_β,(n+1)α

(s)

, n = 0 , 1 , 2 , . . . (38) where it is convenient to utilize χ ˜

ⁿ_β,α

(s) = χ ˜

β,nα

(s) =

_(ξ+s^ξnαβ)^nα

, i.e. to replace α → nα in the expression (35).

We then obtain for the probability for n arrivals within [0 , t ] the expression Φ

_β,α⁽ⁿ⁾

(t ) = ξ

^nα

t

^nαβ

E

_β,(n^nα _αβ+1)

( − ξt

^β

) − ξ

^α

t

^αβ

E

β,((n+1)αβ+1)^(n+1)α

( − ξt

^β

)

. (39)

We refer this distribution to as the ‘generalized fractional Poisson distribution (GFPD)’ [17, 18]. This distri- bution was also obtained by Cahoy and Polito [16]. For α = 1 and 0 < β < 1 the GFPP (39) recovers the fractional Poisson distribution (29) and for α = 1 , β = 1 the standard Poisson distribution (21), and finally for α > 0 and β = 1 the Erlang distribution [17, 18]. For applications in the dynamics in complex systems the asymptotic properties of the GFPD are of interest. For small (dimensionless) times the GFPD behaves as

Φ

_β,α⁽ⁿ⁾

(t ) ≈ (ξt

^β

)

^nα

Γ (n αβ + 1 ) , t ξ

^1β

→ 0 , n = 0 , 1 , 2 , . . . (40)

representing the lowest non-vanishing order in (39). It follows that the GFPD fulfills the initial condition Φ

_β,α⁽ⁿ⁾

(t )

t=0

= δ

_n0

, (41)

reflecting that per construction at t = 0 no event has arrived. Further of interest is the asymptotic behavior

for large (dimensionless) times t ξ

^1β

. To this end, let us expand the Laplace transform for small s → 0 in (38)

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

t

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Φ ( n ) β ,α ( t )

(b) α = 2.0, β = 0.5

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

t

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Φ ( n ) β ,α ( t )

(a) α = 0.5, β = 0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

n

Figure 2: Probability Φ

⁽ⁿ⁾_β,α

(t) as a function of t for different n. (a) α = 0.5 and β = 0.5, (a) α = 2.0 and β = 0.5. In the colorbar we represent n = 1, 2, . . . , 5. The values were obtained numerically using ξ = 1 with Eq. (38). The results for t 1 show the power-law relation Φ

⁽ⁿ⁾_β,α

(t) ∝ t

^nαβ

in Eq. (40).

up to the lowest non-vanishing order in

^s_ξ^β

to arrive at Φ

_β,α⁽ⁿ⁾

(t ) ≈ α

ξ L

⁻¹

{ s

^β−1

} = α Γ (1 − β)

t ξ

^1β−β

, t ξ

^1β

→ ∞ , n = 0 , 1 , . . . , ∞ (42) where this inverse power law holds universally for all α > 0 for 0 < β < 1 and is independent of the number of arrivals n recovering the fractional Poisson distribution for α = 1 of Eq. (33). We notice that for large (dimensionless) observation times an universal (t ξ

^1β

)

^−β

power-law decay occurs which is independent of the arrival number n where α occurs only as a scaling parameter in relation (42). We interpret this behavior as quasi-ergodicity property, i.e. quasi-equal distribution of all ‘states’ n for tξ

^1β

large [17]. The fractional exponent − β further is independent of α thus the power-law is of the same type as in the fractional Poisson process.

In the Figure 2(a) we have plotted the probabilities Φ

_β,α⁽ⁿ⁾

(t ) of Eq. (39) for fixed α and β for different arrival numbers n. One can see that for large times the Φ

_β,α⁽ⁿ⁾

(t ) converge to the same universal behavior independent of n which reflects the asymptotic power-law relation (42). On the other hand the asymptotic power-law behavior for small t is shown in Figure 2(b) (See also relation (40)). The decay to zero lim

t→0

Φ

_β,α⁽ⁿ⁾

(t )

t=0

∼ (ξt

^β

)

^nα

→ 0 (n > 0 ) becomes the more pronounced the larger n. This behavior also can be interpreted that the higher n, the less likely are n arrivals to happen within a small time interval of observation.

The GFPP and the fractional Poisson process for large observation times exhibit the same power-law asymptotic feature (See again asymptotic relation (42)). This behavior reflects the ‘asymptotic universality’

of the fractional Poisson dynamics, the latter was demonstrated in Ref. [33]. The inverse power-law decay occurring for 0 < β < 1 with fat-tailed waiting time PDF indeed is the source of non-Markovian behavior with long-time memory. In the entire admissible range 0 < β ≤ 1 the waiting time PDF χ

_β,α

(t ) and survival probability Φ

_β,α⁽⁰⁾

(t ) maintain their good property of being completely monotonic functions, i.e. they fulfill

⁶

( − 1 )

ⁿ

d

ⁿ

dt

ⁿ

f (t ) ≥ 0 , n = 0 , 1 , 2 , . . . , t > 0 . (43)

6See Ref. [20] for a discussion of this issue for the fractional Poisson process.

An analysis of various aspects of completely monotonic functions is performed in our recent works [8, 34], and see the references therein.

5

CONTINUOUS-TIME RANDOM WALK ON NETWORKS

Having recalled above basic properties of renewal theory

⁷

our goal is now to analyze stochastic motions on undirected networks and lattices that are governed by the GFPP renewal process. To develop our model we employ the continuous-time random walk (CTRW) approach by Montroll and Weiss [19] (and see also the references [2, 21, 22]). In the present section our aim is to develop a CTRW model for undirected networks in order to apply the theory to infinite d -dimensional integer lattices

Z^d

.

We consider an undirected connected network with N nodes which we denote with p = 1 , . . . , N . The topology of the network is described by the positive-semidefinite N × N Laplacian matrix which is defined by [8, 12, 36, 37, 38, 39, 40]

L

_pq

= K

_p

δ

_pq

− A

_pq

(44)

where

A

= (A

_pq

) denotes the adjacency matrix having elements A

_pq

= 1 if a pair pq of nodes is connected and A

pq

= 0 if a pair pq is disconnected. Further we forbid that nodes are connected with themselves thus A

pp

= 0 . In an undirected network adjacency and Laplacian matrices are symmetric. The diagonal elements L

_pp

= K

_p

of the Laplacian matrix are referred to as the degrees of the nodes p counting the number of neighbor nodes of a node p with K

_p

=

PN

q=1

A

_pq

. In order to relate the network topology with random walk features we introduce the one-step transition matrix

W

= (W

_pq

) which is defined by [8, 12]

W

_pq

= 1

K

_p

A

_pq

= δ

_pq

− 1

K

_p

L

_pq

. (45)

Generally, the transition matrix is non-symmetric for networks with variable degrees K

i

6 = K

j

(i 6 = j). The one-step transition matrix W

_pq

defines the conditional probability that a random walker which is on node p jumps in one step to node q where in one step only neighbor nodes with equal probability

_K¹

p

can be reached.

We see in definition (45) that the one-step transition matrix

PN

q=1

W

pq

= 1 and 0 ≤ W

pq

≤ 1 and also the n-step transition matrices

Wⁿ

are (row-)stochastic [8].

Now let us assume that each step of the walker from one to another node is associated with a jump event or arrival in a CTRW with identical transition probability (W

_pq

) for a step from node p to node q. We assume the random walker performs IID random steps at random times 0 ≤ t

₁

, t

₂

, . . . t

_n

, . . . , ∞ in a renewal process with IID waiting times ∆t

k

where the observation starts at t = 0 . To this end let us recall some basic relations holding generally, and then we specify the renewal process to be a GFPP.

Introducing the transition matrix

P(t

) = (P

_ij

(t )) indicating the probability to find the walker at time t on node j under the condition that the walker at t = 0 initially was sitting at node i, we can write [35, 41]

P(t

) =

P(0

)

∞

X

n=0

Φ

⁽ⁿ⁾

(t )W

ⁿ

(46)

where we assume here a general initial condition

P(t

) |

t=0

=

P(0

) which is fulfilled by accounting for the initial conditions Φ

⁽ⁿ⁾

(t )

t=0

= δ

n0

. In this series the Φ

⁽ⁿ⁾

(t ) indicate the probabilities of n (jump-) events in the renewal process, i.e. the probability that the walker performs n steps within [0 , t ] (See Eq. (5)), and (W

ⁿ

)

_ij

indicates the probability that the walker in n jumps moves from the initial node i to node j . We observe in view of relation (13) together with

PN

j=1

(W

ⁿ

)

_ij

= 1 that the normalization condition

PN

j=1

P

ij

(t ) =

P∞

n=0

Φ

⁽ⁿ⁾

(t ) = 1 is fulfilled. The convergence of series (46) can be easily proved by using that

W

has uniquely eigenvalues | λ

m

| ≤ 1 and with | χ(s) ˜ | ≤ 1 [8, 17]. Let us assume that at t = 0 the walker is sitting on departure node i thus the initial condition is given by P

ij

(0 ) = (δ

ij

), then the Laplace transform of (46) writes [17]

P(s) =

˜ (1 − χ(s)) ˜

s {

1

− χ(s)W ˜ }

⁻¹

(47)

7For further details on renewal theory, see e.g. [35].

where

P(s) has the eigenvalues [17]

˜

P(m, ˜ s) = (1 − χ(s)) ˜ s

1

(1 − λ

_m

χ(s)) ˜ , m = 1 , . . . , N . (48)

The λ

m

indicate the eigenvalues of the one-step transition matrix

W. This expression is the celebrated

Montroll-Weiss formula [19] and occurs in various contexts of physics.

6

GENERALIZED SPACE-TIME FRACTIONAL DIFFUSION INZ^d

In this section our aim is to develop a CTRW which is a random walk subordinated to a GFPP. For the random walk on the network we allow long-range jumps which can be described when we replace the Laplacian matrix by its fractional power in the one-step transition matrix (45). In this way, the walker cannot only jump to connected neighbor nodes, but also to far distant nodes in the network [5, 6, 8, 34, 36, 37, 38, 39, 42, 43]. The model to be developed in this section involves both space- and time-fractional calculus. As an example we consider the infinite d -dimensional integer lattice

Z^d

. The lattice points

p

= (p

₁

, . . . , p

_d

) (p

_j

∈

Z0

) represent the nodes where we assume each node is connected to any of its 2d neighbor nodes. The

Z^d

is an infinite cubic primitive d -dimensional lattice with lattice-constant one. In this network any node has identical degree 2d. The one-step transition matrix with the elements W

^(µ)

(p −

q) has then the canonic representation [8, 38]

W

^(µ)

(p −

q) =

1 (2 π)

^d

Z π

−π

dk

1

· · ·

Z π

−π

dk

_d

e

^ik·(p−q)

λ

^(µ)

(k) (49)

with the eigenvalues

λ

^(µ)

(k) = 1 − 1

K

^(µ)

η

^µ2

(k), η(k) = 2d − 2

d

X

j=1

cos(k

_j

), 0 < µ ≤ 2 (50) where

k

= (k

1

, . . . , k

d

) denotes the wave-vector with − π ≤ k

j

≤ π. One can show that the fractional index µ is restricted to the interval 0 < µ ≤ 2 as a requirement for stochasticity of the one-step transition matrix [8, 34, 38]. In (50) the constant K

^(µ)

can be conceived as a fractional generalization of the degree and is given by the trace of the fractional power of Laplacian matrix, namely [8, 34]

K

^(µ)

= 1

N tr (L

^µ2

) = 1 N

N

X

m=1

(η

m

)

^µ2

(51)

where η

m

denote the eigenvalues of the Laplacian matrix (44) and in an infinite network the sum in (51) has to be performed in the limit N → ∞ . In the

Z^d

the fractional degree with Eq. (51) is then determined from [8, 34]

K

^(µ)

= 1 (2 π)

^d

Z π

−π

dk

1

· · ·

Z π

−π

dk

_d

(η(k))

^µ2

(52)

with the eigenvalues given in Eq. (50). It is necessary to account for the fractional degree since it plays the role of a normalization factor in the one-step transition matrix (See Eq. (45)).

For the present analysis it is sufficient to consider K

^(µ)

as a (positive) constant where for µ = 2 recovers K

^(µ=2)

= 2d the degree of any node. The transition matrix (46) is then determined by its Laplace transform (47) which writes in the

Z^d

as

P ˜

^(µ)

(p −

q,

s) = (1 − χ(s)) ˜ s

1 (2 π)

^d

Z π

−π

dk

1

. . .

Z π

−π

dk

d

P ˜ (k, 0 ) e

^ik·(p−q)

(1 − λ

^(µ)

(k)˜ χ(s)) (53) where P ˜ (k, 0 ) indicates the Fourier transform of the initial condition which has the Fourier representation

P

^(µ)

(p −

q,

t = 0 ) = P

0

(p −

q) =

1 (2 π)

^d

Z π

−π

dk

1

· · ·

Z π

−π

dk

_d

e

^ik·(p−q)

P ˜ (k, 0 ). (54)

In order to analyze the diffusive limit of (53), i.e. its long-wave approximation it will be sufficient to account for the eigenvalues (50) for k → 0 (where k = |

k

| ). Then we have with η

^µ2

(k) ≈ k

^µ

the behavior

λ

^(µ)

(k) ≈ 1 − 1

K

^(µ)

k

^µ

, 0 < µ ≤ 2 , k → 0 . (55)

These equations hold so far for space-fractional walks for an arbitrary renewal process with waiting time PDF χ(t ) = L

⁻¹

{ χ(s) ˜ } .

Now let us consider a space-fractional walk subordinated to a GFPP with χ(s) = ˜ χ ˜

_β,α

(s) of Eq. (34).

We denote the corresponding transition matrix of this stochastic motion as

P^(µ)_β,α

(t ) which contains three index parameters 0 < µ ≤ 2 , 0 < β ≤ 1 and α > 0 and one time scale parameter ξ (of units sec

^−β

).

In order to derive the generalized space-time fractional diffusion equation it is convenient to proceed in the Fourier-Laplace domain. The Fourier-Laplace transform of the

P^(µ)_β,α

(t) is then with Eq. (53) given by the Montroll-Weiss equation

P ˜

_β,α^(µ)

(k, s) = ˜ P(k, 0) (1 − χ ˜

_β,α

(s)) s

1

(1 − χ ˜

_β,α

(s)λ

^(µ)

(k)) = ˜ P (k, 0) s

⁻¹

1 +

₍₁₋^χ˜β,α_χ˜ ^(s)

β,α(s)) η^µ2(k)

K^(µ)

(56)

≈ P(k, ˜ 0) s

⁻¹

1 +

₍₁₋^χ˜β,α_χ˜ ^(s)

β,α(s)) k^µ K^(µ)

, k → 0 (57) containing also the Fourier transform P ˜ (k, 0 ) of the initial condition (54). The exact equation (56) can be rewritten as

− ξ

^α

η

^µ2

(k)

K

^(µ)

P ˜

_β,α^(µ)

(k, s) = (s

^β

+ ξ)

^α

P ˜

_β,α^(µ)

(k, s) − ξ

^α

P ˜

_β,α^(µ)

(k, s) + ξ

^α

− (s

^β

+ ξ)

^α

s

P(k, ˜ 0 ), k

_j

∈ [ − π, π]. (58) Transforming back this equation into the causal time domain and by using Eqs. (54) and (53) yields the generalized time-fractional matrix equation

− ξ

^α

K

^(µ)L^µ2

·

P^(µ)_(β,α)

(t ) =

₀

D

^β,αt

·

P^(µ)_β,α

(t ) − ξ

^αP^(µ)_β,α

(t ) +

P₀n

ξ

^α

Θ(t ) − K

_β,α⁽⁰⁾

(t )

o

, t ≥ 0 (59)

which we refer to as ‘generalized space-time fractional Kolmogorov-Feller equation’ where 0 < µ ≤ 2 with

0 < β ≤ 1 and α > 0 . This equation was obtained and analyzed recently for normal walks (µ = 2 )

subordinated to a GFPP [17, 18]. Equations of the type (59) generally describe the generalized space-

time fractional diffusion on undirected networks connecting the network topology (contained in Laplacian

matrix

L) with the GFPP-governed stochastic motion on the network. We used notation L^µ2

which denotes

the fractional power of Laplacian matrix

L, and P^(µ)_(β,α)

(t ) the transition matrix with the initial condition

P^(µ)_(β,α)

(t = 0 ) =

P0

where all these matrices are defined in

Z^d

. Since the

Z^d

is an infinite network, these are

symmetric and circulant ∞ × ∞ matrices with elements L

^(µ)

(p −

q),

P

_(β,α)^(µ)

(p −

q,

t ),

P0

(p −

q), respectively,

where

p,q

∈

Z^d

. In Eq. (59) we have introduced the causal convolution operator

0

D

^β,αt

and the causal

function K

_β,α⁽⁰⁾

(t ) which were obtained in explicit forms [17, 18]

D

^β,α

(t ) = L

⁻¹n

(s

^β

+ ξ)

^α o

= d

^dαβe

dt

^dαβe

Θ(t )d

^β,α

(t )

= d

^dαβe

dt

^dαβe

(Θ(t)d

^β,α

(t − τ )) =















d^dαβe dt^dαβe

Θ(t)t

^{dαβe−βα−1}P∞

m=0 α!

(α−m)!m!

(ξt^β)^m Γ(βm+dαβe−βα)

, αβ / ∈

N

d

^αβ

dt

^αβ

δ(t ) + Θ(t ) d dt

∞

X

m=0

α!

(α − m )!m !

(ξt

^β

)

^m

Γ (mβ + 1 )

!

, αβ ∈

N

=







d^dαβe

dt^dαβe

Θ(t)t

^{dαβe−βα−1}

E

α,β,(dαβe−αβ)

(ξt

^β

)

, αβ / ∈

N

d^αβ

dt^αβ

δ(t) + Θ(t)

_dt^d

E

_α,β,1

(ξt

^β

)

, αβ ∈

N

.

(60) In these expressions we introduced the ceiling function d γ e indicating the smallest integer greater or equal to γ and the function E

_c,a,b

(z) = E

_a,b^−c

( − z) where E

_v,w^u

(ζ) indicating the Prabhakar-Mittag-Leffler function (36).

The operator

0

D

^β,αt

acts on a causal distribution P(t ) such as in Eq. (59) in the following way

0

D

t^β,α

· P (t ) = d

^dαβe

dt

^dαβe

Z t 0

d

^β,α

(t − τ )P (τ )dτ. (61)

The function K

_β,α⁽⁰⁾

(t ) of equation (59) was obtained as

K

_β,α⁽⁰⁾

(t ) = L

⁻¹

(s

^β

+ ξ)

^α

s

=



















Θ(t)t

^−αβ

E

α,β,1−αβ

(ξt

^β

), 0 < αβ < 1 d

^dαβe−1

dt

^dαβe−1

Θ (t )t

^{dαβe−βα−1}

E

α,β,(dαβe−αβ)

(ξt

^β

)

αβ > 1 , αβ / ∈

N

d

^αβ−1

dt

^αβ−1

δ(t ) + Θ(t) d

dt E

_α,β,1

(ξt

^β

)

, αβ ≥ 1 ∈

N

.

(62) Equation (59) governs the ‘microscopic’ stochastic motions of the space fractional walk subordinated to a GFPP.

7

DIFFUSION-LIMIT

Our goal now is to determine the ‘diffusion-limit’ of above stochastic motion to obtain a ‘macroscopic picture’

on spatial scales large compared to the lattice constant 1 of the

Z^d

. To this end it is sufficient to consider Montroll-Weiss equation (56) for k small. Then Eq. (58) can be rewritten as

− ξ

^α

k

^µ

K

^(µ)

P ˜

_β,α^(µ)

(k, s) ≈ (s

^β

+ ξ)

^α

P ˜

_β,α^(µ)

(k, s ) − ξ

^α

P ˜

_β,α^(µ)

(k, s) + ξ

^α

− (s

^β

+ ξ)

^α

s P ˜ (k, 0 ), k → 0 . (63) In order to derive the ‘diffusive limit’ which corresponds to the space-time representation of this equation, it appears instructive to consider the long-wave contribution of some kernels such as the fractional power of the Laplacian matrix in

Z^d

. The fractional Laplacian matrix in

Z^d

has the canonic form [8, 38]

[L

^µ2

]

p−q

= 1 (2 π)

^d

Z π

−π

dk

1

· · ·

Z π

−π

dk

d

e

^ik·(p−q)

η

^µ2

(k), 0 < µ ≤ 2 (64)

where η(k) are the eigenvalues of the Laplacian matrix in

Z^d

defined by Eq. (50). Let us consider the

contribution generated by small k → 0 by integrating over a small d -cube | k

j

| ≤ k

c

1 around the origin